Beyond Platonic: How to Build Metal–Organic Polyhedra Capable of Binding Low-Symmetry, Information-Rich Molecular Cargoes

The field of metallosupramolecular chemistry has advanced rapidly in recent years. Much work in this area has focused on the formation of hollow self-assembled metal-organic architectures and exploration of the applications of their confined nanospaces. These discrete, soluble structures incorporate metal ions as ‘glue’ to link organic ligands together into polyhedra.Most of the architectures employed thus far have been highly symmetrical, as these have been the easiest to prepare. Such high-symmetry structures contain pseudospherical cavities, and so typically bind roughly spherical guests. Biomolecules and high-value synthetic compounds are rarely isotropic, highly-symmetrical species. To bind, sense, separate, and transform such substrates, new, lower-symmetry, metal-organic cages are needed. Herein we summarize recent approaches, which taken together form the first draft of a handbook for the design of higher-complexity, lower-symmetry, self-assembled metal-organic architectures.

The field of metallosupramolecular chemistry has advanced rapidly in recent years. Much work in this area has focused on the formation of hollow self-assembled metal−organic architectures and exploration of the applications of their confined nanospaces. 1−5 These discrete, soluble structures incorporate metal ions as "glue" to link organic ligands together into polyhedra. Their hollows have found applications in binding and sensing guests, 6−8 stabilizing reactive molecules, 9−13 and catalyzing reactions as enzymes do. 14−19 Most of the architectures employed to date have been highly symmetrical, as these have been the easiest to prepare (Figure 1). 20 An understanding of the design principles underpinning the formation of high-symmetry metal−organic cages, 1 such as tetrahedra, 21−24 cubes, 25−28 and octahedra, 29−34 has enabled their synthesis and application. 35−38 Modification of these structures, either before or after assembly, 39−41 can imbue them with new functions. 42 Such functions include modulation of the guest-binding properties, 43−46 phase transfer (whereby a capsule and its cargo are induced to move between phases), 47,48 and enabling the formation of higher-order metal−organic cagebased materials. 49−56 Such high-symmetry structures contain pseudospherical cavities and thus bind roughly spherical guests optimally, 25,35−38 although asymmetric guests can also be encapsulated. 10,12,20,57−59 In some cases more than one smaller guest is bound within a relatively large cavity, 57 or the flexibility of a guest enables it to adopt a folded conformation with a complementary size and shape for the cage cavity. 18,59,60 Biomolecules and high-value synthetic compounds are rarely isotropic, highly symmetrical species. 61 To bind, sense, separate, and transform such substrates, new lower-symmetry metal− organic cages are needed. In response to this need, recent work has focused upon the construction of metal−organic cages with interior cavities of reduced symmetry.
Many early examples of lower-symmetry structures were discovered serendipitously. Only a limited number of structure types beyond the Platonic solids were prepared using established design principles. The great promise of lower-symmetry structures to bind lower-symmetry guests selectively ( Figure  1) has motivated efforts to decipher the rules underpinning the formation of complex architectures. 62−64 Herein we outline different approaches that taken together form the first draft of a handbook for the design of higher-complexity, lower-symmetry, self-assembled metal−organic architectures.

Classification of Approaches
The design of metal−organic architectures has been discussed in terms of the following four strategies: the directional-bonding approach, 1 the symmetry-interaction approach, 65 the molecularpaneling approach, 66 and the weak-link approach. 67−69 Each of these strategies has been employed to form metallomacrocycles or high-symmetry three-dimensional architectures, often with Platonic geometries. With careful consideration, these design strategies can also be employed to form lower-symmetry structures that deviate from the Platonic solids. However, in this review we have opted for a method of classification that deviates from the strategies noted above because approaches enabling the formation of more complex metal−organic assemblies have recently been established that do not neatly fall within these categories. We focus instead upon the properties of the building blocks along with reaction conditions. This organization lends itself to the aim of this reviewto act as a preliminary guide for the further design of complex selfassembled architectures. Examples of coordination cages with structures corresponding to Platonic solids, which are well-adapted to pseudospherical guests, contrasted with more complex "beyond Platonic" cages, which are primed for binding of anisotropic guests. 25,35,36,59,62−64 Using this building block/reaction condition-based classification, we have identified six broad categories of approach ( Heteroleptic architectures incorporate multiple different ligands (Figure 2a). A particular challenge in this approach is to ensure that the different building blocks integrate into a single product rather than segregating to form simpler structures, each containing only one type of building block. One strategy developed to overcome this challenge involves harnessing the enthalpic and entropic driving forces that govern self-assembly in order to favor a heteroleptic structure.
A similarly intuitive approach involves the use of ligands with greater structural complexity or reduced symmetry, which then translates to the assembly of more complex three-dimensional architectures (Figure 2b).
Flexibility is often incorporated within ligands by the addition of alkyl spacers. Such enhanced flexibility can increase the array of feasible structures in comparison with the use of more rigid ligands, but it can also decrease the predictability of the selfassembly process (Figure 2c).
Complexity based upon solvent, anion, and template effects relies upon altering the self-assembly reaction conditions in order to favor structural complexity (Figure 2d). This method is well-established for producing complex metal−organic architectures. However, as with enhancing ligand flexibility, predicting the outcome of self-assembly using this approach can be challenging.
Multimetallic architectures either contain more than one type of metal center or have vertices that consist of homometallic clusters. Both cases can introduce coordinational flexibility, enabling the formation of architectures with increased structural complexity (Figure 2e).
The sixth approach to generating complex structures in a controlled and predictable manner is the incorporation of geometric constraints into the ligands. These geometric constraints can act to frustrate the formation of simpler structures, thus favoring the construction of architectures with greater complexity (Figure 2f). Examples in which steric control or non-covalent interactions are used to form complex metal− organic structures are also highlighted in this section.

Scope of the Review
This review focuses on techniques used to prepare metal− organic architectures by self-assembly of organic ligands and metal ions. Some complex structures that form with metalcluster cores or with metal clusters as vertices 70−72 are also included. The term "complex structure" within this review generally refers to structures that deviate from a framework corresponding to one of the high-symmetry Platonic or Archimedean solids. Some examples of structures that outwardly resemble these simple polyhedra, but with reduced symmetry, are included, particularly when the source of the reduced symmetry can be determined.
Although a key motivation for this review is to aid those who might wish to design new lower-symmetry structures for new applications, we focus on construction principles as opposed to the utility and functions of these structures. As the field that we attempt to cover is wide-ranging and fast-moving, omissions in our coverage will be inevitable. We apologize for these in advance.
of unusual architectures with complex geometries. In practice, however, achieving the selective formation of a single structure from a range of possibilities can be challenging. This section explores ways in which this challenge has been overcome, focusing on approaches that may allow general routes to heteroleptic structures.

Heteroleptic Selectivity by Destabilization of Homoleptic Assemblies
The selective assembly of a single heteroleptic metal−organic architecture is often entropically disfavored. For example, a square-planar metal vertex coordinated by 2 equiv of two different ligands through monodentate donors (i.e., ML 1 2 L 2 2 ) may coexist with other mixed-ligand (i.e., ML 1 1 L 2 3 , ML 1 3 L 2 1 ) and homoleptic (ML 1 4 , ML 2 4 ) vertices. One way to overcome this tendency is to build in an enthalpic driving force for heteroleptic assembly. Stang et al. found that the principle of charge separation could drive the assembly of less-symmetric structures. 73 This approach, shown in Figure 3, depends on the use of platinum(II) centers with two strong-field ligands in a cis configuration and both pyridine (1) and carboxylate (2) donor ligands. After coordination of a pyridine donor to platinum, the pyridine nitrogen atom bears a partial positive charge. When two pyridine donors are adjacent, they repel each other electrostatically ( Figure 3, 3). This repulsion is ameliorated when one of the pyridine donors is replaced by a carboxylate (Figure 3, 5). This reduction in repulsion thus leads to the observed preference for heteroleptic coordination.
The Stang group has employed this concept extensively, for example to form an array of trigonal, tetragonal, and hexagonal prisms and other heteroleptic complexes, by combining cis-Pt II (PEt 3 ) 2 (OTf) 2 with 1,4-benzenedicarboxylate (6) ( Figure  3) and three-, four-, or sixfold-symmetric pyridine donors. 73−75 Figure 3 shows the structures of 6, threefold-symmetric donor 7, and the self-assembly product 8. In collaboration with the Huang group, this concept was extended to generate highly emissive platinum(II) metallacages using a fourfold-symmetric pyridine donor component that contains a fluorophore that undergoes aggregation-induced emission. 76 The strict spatial separation enforced by the metal−organic architecture preserved the fluorescence in both high-and low-concentration regimes, allowing white-light emission. Similar principles were recently reported in a metallacycle where a high degree of intramolecular twist constrained the incorporated anthracenes, increasing the emission intensity. 77 Furthermore, the same group, working with the Sun group, showed that metal−organic capsules can self-assemble into soft superstructures of up to the millimeter scale. 78 Combinations of nitrogen-donor and carboxylate ligands have also been used to create molecular rectangles based on palladium. 79 The formation of cages containing perylene diimide panels, which can bind polycyclic aromatic hydrocarbons, was recently reported by Zhang et al. 80 By combining the orange emission of the cage and blue emission of a captured guest, white-light emission was obtained. Differences in fluorescence quantum yield between the solid-state and solution were also exploited to create hidden messages that were revealed upon exposure of the system to acetonitrile vapor. 80 As shown in Figure 4, Severin et al. reported that strained homoleptic assemblies such as 9 rearrange following the addition of an extra ligand. 81 Metallomacrocycle 9 is strained, and its strain is alleviated in heteroleptic assembly 11, thus providing a driving force to counter the entropic cost of integrating more building blocks. In homoleptic assembly 9, one carboxylate at each metal center forms a four-membered chelate ring, the strain of which is relieved as these carboxylates become monodentate in flexible trigonal prism 11 following the addition of 2,4,6-tris(pyridin-4-yl)-1,3,5-triazine (10). The resulting monodentate binding endows product 11 with a high degree of flexibility. In the absence of a guest, the trigonal-prismatic framework of 11 collapses in the solid state, forming a compressed structure without an interior cavity. However, when coronene is added, the trigonal prism expands to encapsulate two coronene guests in the solid state. This work shows that flexible coordination cage cavities can be generated  not just from flexible organic ligands but also from coordinational flexibility about metal centers.
Schmittel and co-workers reported the application of their "heteroleptic terpyridine and phenanthroline metal complexes" (HETTAP) concept to generate myriad of self-assembled structures, including nanoprisms. This approach relies on steric hindrance around the phenanthroline units to prevent homoleptic assembly. 82,83 By combining a threefold-symmetric bulky phenanthroline-based ligand (12) with linkers of different lengths (i.e., 13, shown in Figure 5), the authors generated a series of trigonal prisms of differing heights of the general form Cu I 6 L 1 2 L 2 3 . The heteroleptic architecture of 14 was further stabilized, eliminating minor byproducts, by the addition of a suitable bridging guest capable of coordinating between the zinc centers in the porphyrins of the ditopic ligands. A planar tridentate pyridine ligand that binds in the central belt of the three porphyrins drives the quantitative formation of the heteroleptic structure. Similar approaches, heteroleptic bis(phenanthroline) complexation (HETPHEN) and heteroleptic pyridine and phenanthroline complexation (HETPYP), have also been shown to selectively yield heteroleptic metal−organic complexes. 83 A system may be guided toward heteroleptic assembly through destabilization of alternative homoleptic products that would undergo steric clash. An early seminal example was provided by Yoshizawa and co-workers, who combined sterically hindered and unhindered ligands containing two pyridines to form heteroleptic trigonal prisms. 84 Similar approaches have been taken more recently by the Clever group, who used steric bulk to destabilize certain assemblies in order to favor heteroleptic species. 85,86 We developed this concept during the selective formation of a copper(I) rhomboidal diporphyrin prism, shown in Figure 6. 87 Upon mixing of 8 equiv of the bis(diphenylphosphino)benzene (15) struts, 8 equiv of 2formylpyridine (16), a guest (17), and 2 equiv of the tetratopic zinc(II)porphyrin (18) with 8 equiv copper(I), rhomboidal prism 19 forms. The offset between the porphyrins within 19 leads to its selective binding of 3,3′-bipyridine (17) between zinc centers.
The formation of a homoleptic L 2 Cu I 4 porphyrin copper(I) sandwich complex is disfavored by steric clashes between the phenyl groups, and the formation of copper(I) complexes involving the coordination of more than two phosphines is disfavored by the steric bulk of the phenyl groups on phosphorus. The simplest assembly that gives coordinatively saturated copper(I) is thus prism 19. The preference for heteroleptic assembly is likely reinforced by the known preference for copper(I) to selectively form mixed phosphine−pyridine complexes. 88 The strategy of using ligands with donors of differing coordinative strength can also drive heteroleptic assembly in concert with the steric effects noted above. As shown in Figure 7, Lehn and co-workers reported a series of cylindrical complexes based on combinations of linear oligo(bipyridine) ligands such as 20 with planar, threefold-symmetric hexaazatriphenylene (HAT) ligand 21 and either silver(I) or copper(I). 89,90 The electron-deficient HAT ligands bind less strongly than  bipyridines, and their phenyl groups generate steric clashes when two HAT ligands bind around a single metal ion. Assemblies formed from HAT 21 alone would thus be relatively unstable as well as polymeric in nature and thus entropically less favored than the discrete cylindrical assemblies that are observed to form. Lehn et al. used linear ligands containing up to four bipyridine motifs, thus generating cylinders with up to three spatially separated binding pockets. Although the host−guest behavior of this system was not investigated in detail, the principle of using spatially separated binding pockets within the same assembly was further explored by others, such as Clever 91,92 and Crowley 93 (see Figure 63 in section 7.4).

Ligand Shape Complementarity
Clever et al. reported a multitude of different heteroleptic Pd II 2 L 4 lantern-type structures, based upon their initial work with analogous homoleptic structures, that contain bidentate ligands incorporating pyridine donors with parallel coordination vectors. 94,95 Clever's approach to forming heteroleptic structures exemplifies the use of shape complementarity. 96,97 In the example in Figure 8, bidentate ligand 23 contains isoquinoline donors, and another, 25, contains pyridine donors, each with nonparallel coordination vectors. 96 Strain is thus incorporated into homoleptic structures 24 and 26, as the offset coordination vectors cannot close up into a polyhedron by coordinating to square-planar palladium(II) without distortion. When mixed, however, the two ligands come together to form Pd II 2 23 2 25 2 heteroleptic architecture 27 in which each ligand is cis to its complementary partner, thus forming a tilted lantern architecture. The extension of this concept to a wider variety of ligands subsequently enabled the discovery of an unusual selfpenetrating heteroleptic cage architecture. 98 Clever and coworkers reported a range of interpenetrated and heteroleptic systems based on similar principles. 99−103 Severin and coworkers recently reported the use of similar "banana-shaped" ligands to create heteroleptic cages based on a virtual combinatorial library involving six separate ligands. This led to the formation of a trigonal-antiprismatic [Pd II 6 L 6 L′ 6 ](BF 4 ) 12 structure. 104 The Fujita group reported the assembly of a heteroleptic cantellated tetrahedron from ligands 28 and 29 (Figure 9). 105 These ligands have the same angle between coordinating groups but different lengths. Each ligand forms a Pd II 12 L 24 cuboctahedral assembly when combined with Pd II on its own. However, when combined in a 1:1 ratio, the two diastereomers of product 30 shown in Figure 9 form instead. Rather than narcissistic selfsorting, where each homoleptic assembly forms independently, or random mixing, where a collection of different assemblies form with different ratios of the two ligands incorporated, the system instead produces only Pd II 12 28 12 29 12 assemblies. Each cis pair of ligands coordinating the same Pd II forms part of a smaller Pd II 3 28 3 or larger Pd II 3 29 3 triangular metallomacrocyle, with four of each of these metallomacrocycles covering the cage surface, sharing edges with Pd II 4 28 2 29 2 rectangles. The Pd II 12 28 12 29 12 constitution of 30, as opposed to other ratios of 28 to 29, thus minimizes strain among these triangles and rectangles. Similar principles were used by Benkhaüser and Luẗzen to create a dinuclear copper(I) molecular kite from subcompo-  nents that did not assemble into discrete, unstrained structures individually. 106

Entropy as a Driving Force for Heteroleptic Assembly
We recently reported a system that undergoes heteroleptic assembly by entropically favoring the mixed architecture ( Figure  10). 62 Cubes 36 and tetrahedra 34 or 35 are in equilibrium with triangular prisms 37 or 38, respectively. The triangular-prismatic architecture is disfavored enthalpically, but its formation is favored entropically for two reasons. First, the triangular prism has a greater number of conformational microstates: each porphyrin unit adopts a saddled configuration, bowing in or out, in the triangular prism, whereas the porphyrins must lie planar in the cube. Second, the combined cavity volume of triangular prisms 37 or 38 is smaller than the combined volumes of the corresponding cubes and tetrahedra. Thus, fewer solvent molecules are trapped inside the cavities of triangular prisms 37 or 38 relative to the tetrahedra (34 or 35) and cube (36), leading to a more favorable entropy.  Homoleptic structures, such as 34, 35, and 36, have higher symmetry and more-spherical cavities than the corresponding heteroleptic structures 37 and 38. Such spherical, isotropic cavities are poorly adapted to the binding of more complex, anisotropic molecules of biological interest. A key advantage of the less-symmetric heteroleptic architectures 37 and 38 is the ability to bind higher-value, more complex substrates (e.g., 39− 44; Figure 10c) than the more symmetric homoleptic structures.

Favorable Interactions between Ligands to Drive Heteroleptic Assembly
Heteroleptic assembly can be favored by engineering of additional favorable interactions that are not present in the corresponding homoleptic systems. We reported a system of mixed pyrene-and naphthalenediimide-based pyridylimine ligands ( Figure 11). 107 Alone, each ligand forms a stable homoleptic structure. However, together subcomponents 45 and 46 form Fe II 4 45 4 46 2 elongated structure 47, which has a different connectivity than either of the homoleptic assemblies. Differentially substituted subcomponent 48, when combined with 46, forms the original homoleptic architectures in an example of narcissistic self-sorting. The selective formation of heteroleptic structure 47 is driven by favorable aromatic stacking interactions between electron-rich and electron-deficient aromatic units that exist only in the mixed architecture. This stacking drives the assembly of the mixed architecture even in the presence of a guest that binds to only one of the possible homoleptic species. This system shows the importance of aromatic stacking interactions in metal−organic architectures.
Such stacking interactions were also critical in driving the formation of a recently reported twisted trigonal-prismatic architecture. 108 Jung and co-workers also reported a catenated architecture based on the stacking of electron-deficient and electron-rich aromatic rings. 109 In a similar vein, Yuasa et al. demonstrated that favorable interligand charge-transfer interactions can cause a preference for heteroleptic assemblies over homoleptic alternatives. 110 Fujita and co-workers developed a heteroleptic Pt II 6 L 2 L′ 3 trigonal prism whose formation is templated by a rigid, flat aromatic guest that binds only in the heteroleptic architecture. Guest binding thus drives selective formation of the heteroleptic trigonal prism. After formation, the guest can be removed by extraction with an apolar solvent, leaving the empty trigonal prism. 111 The cavity thus formed can then be used to stabilize the pairing of DNA nucleobases in aqueous solution. 63

Complementary Binding Sites
Stang and co-workers have made extensive use of the squareplanar geometric preference of palladium(II) and platinum(II) centers to construct metal−organic assemblies. 112 They have obtained heteroleptic assemblies using the concept of complementary binding sites, whereby each component is unable to self-assemble without a complementary partner. As shown in Figure 12, cuboctahedron 53 can be prepared by the assembly of threefold-symmetric, planar metalloligand 51 with bidentate pyridine donor 52. As the metal centers are covalently integrated into one ligand, a second ligand is required for assembly into the nanometer-scale product 53. This work was subsequently extended to form similar chiral adamantanoid cages that incorporate optically active building blocks. 113 Similar principles were previously used by Bosnich and coworkers to selectively generate platinum(II)-based heteroleptic rectangles using terpyridine and monopyridine ligands. 114 The Nabeshima 115 and Yam 116 groups also used this concept to create molecular rectangles, and the area of complementary ligand denticity has recently been reviewed. 117 The advantages of combining different donor groups in the same system were further established by Mukherjee and co-workers, who formed open "swings" and "boats" by using pyridine donors in combination with imidazole donors. 118 These structures can bind C 60 and catalyze Knoevenagel condensations. 119 Other groups have further developed the concepts described above to form heteroleptic cages with useful properties. For example, the groups of Ribas, Costas, and Reek reported the formation of a tetragonal-prismatic supramolecular cage from the combination of tetratopic metalloporphyrin tetracarboxylate 55 and macrocycle 54 containing two palladium(II) centers, each coordinated by three nitrogen donors ( Figure 13). 120 In this system, the coordination preferences of Pd II are satisfied by one carboxylate ligand and one macrocyclic ligand, leading to the formation of structure 56 with Pd II 8 54 4 55 2 composition. This structure encapsulates aminophosphite ligand 57, which coordinates rhodium to form 58. The active supramolecular catalyst thus formed (59) operates with a greater degree of chiral induction due to cage control over the second coordination sphere. Similar capsules have been reported and used for the selective extraction and functionalization of fullerenes. 121−123 In collaboration with the von Delius group, the Ribas group recently reported the formation of a "matryoshka" Russian dolltype assembly and its application in the selective formation of a single trans-3 fullerene bisadduct. 124 Jin and co-workers reported a system of heteroleptic cages where selective assembly is driven by the interplay between two pairs of distinct chelating sites, a harder O,O′ site and a softer N,N′ site, on a single hydroxamate ligand (60), as shown in Figure 14. 125 Half-sandwich iridium and rhodium metal centers assemble with auxiliary pyridine-based ligands, such as 4,4′bipyridine (61), to form tetragonal and trigonal prisms. The D 2symmetric diastereomer of cage 62 ( Figure 14) binds triflate as a guest and template.
The hard/soft character of ligand 60 was also used to form heterometallic macrocycles with palladium and iridium centers selectively incorporated into the same framework. Within these heterometallic structures, palladium binds the softer nitrogen donors, whereas iridium binds the harder oxygen donors. One of these macrocycles encapsulated tetrathiafulvalene between parallel hydroxamate ligands. 125 The authors recently reported an extension of this system in which symmetric bipyridine 61 is replaced by a bridging unit containing one pyridine and one carboxylate donor site, forming a D 2 -symmetric heteroleptic species selectively. 126 We reported a system of Pd II -based macrocycles and cages whose assembly is controlled by the addition of appropriate pyridine-containing templates to the assembled Pd II -bound macrocycles. Each Pd II is coordinated by three nitrogens from the macrocycle ( Figure 15) and one from the bridging ligand. 127 The subcomponents 2,6-diformylpyridine (63) and flexible dianiline 64 assemble around palladium(II) templates to generate metal−organic macrocycles containing either three or four Pd II centers, depending on the tri-or tetratopic nature of the pyridine template used. When we employed linear, ditopic pyridine template 65, which has a geometry ill-adapted to incorporation within a single macrocycle, three-dimensional capsule 66 was generated. This structure ( Figure 15) includes a trimeric macrocycle at each end with bridging ligands 65 between them. Assembly 66 forms cooperatively, with no structures observed containing fewer than three bridging ligands. Structure 66 encloses a small cavity, which was found to bind tetrafluoroborate selectively.
Similar systems were extended to form truncated tetrahedra and other metal−organic cages by the use of a tritopic aniline ligand. The dynamic pyridylimine bonds formed during selfassembly could be cleanly reduced to form secondary amines, thus disabling the equilibration process and fixing the structures formed.

Kinetic Traps
Crowley and co-workers reported a novel approach to generating heteroleptic architectures that employs kinetic traps rather than favoring a thermodynamic product ( Figure  16). 128 Pd II 2 L 4 lantern architecture 69, formed from bidentate   pyridine-containing ligand 67 with parallel coordination vectors ( Figure 16), is combined with another ligand 68 containing 2aminopyridines. Ligand 68 forms stronger bonds to palladium, so thermodynamics favors its incorporation. When excess ligand 68 is added to Pd II 2 67 4 lantern 69, Pd II 2 67 2 68 2 lantern 71 forms selectively in a cis configuration. The selectivity for the cis isomer is attributed to hydrogen bonding between adjacent amino groups. The selective formation of a Pd II 2 67 2 68 2 lantern, rather than complete substitution to form a homoleptic Pd II 2 68 4 structure, is attributed to the effects of steric repulsion between the 2-amino groups and incoming pyridine ligands in the proposed associative mechanism. This repulsion increases the energetic barrier to ligand exchange, enabling the selective formation of the heteroleptic species. Calculations suggested that the heteroleptic species is a kinetically trapped metastable species rather than the thermodynamic product, and competition experiments supported this idea. Hydrogen bonding between the pyridine α-CH and adjacent 2-aminopyridine groups is inferred to reinforce this kinetic stability.
An intriguing use of kinetic control in self-assembly was reported by Lusby, Barran, and co-workers, who used the low lability of cyclometalated platinum corners to create trigonalprismatic assemblies ( Figure 17). 129 The identity of the product depends on the sequence of addition rather than the thermodynamic stability of the product. Starting from a platinum complex with one pyridine, one dimethyl sulfoxide, and two phenylato ligands, a bi-or terpyridine ligand is then added. This additional ligand displaces weakly bound dimethyl sulfoxide to form an intermediate complex with either twofold (72) or threefold (74) symmetry. In the case of twofold-symmetric intermediate 72, tritopic pyridine ligand 10 is then added, which forms a new coordination bond trans to a phenylato ligand. This process displaces another phenylato ligand, which is then protonated. The phenyl group thus released is left above the threefold-symmetric face of trigonal prism 73.
If instead bipyridine 61 is added to threefold-symmetric intermediate 74, the released phenyl groups of product 75 instead stack above the twofold-symmetric ligand. This isomerism is further manifested in the mass spectrometry data collected, where the weaker coordination bonds trans to the phenylato group are observed to rupture preferentially. This approach provides an example of how the sequence of addition can control the outcome of a self-assembly process and thus provides a novel mode of generating structural complexity.
This section has reviewed different approaches for generating heteroleptic structures, which frequently have novel, lowersymmetry architectures. We have explored how control over both the entropy and enthalpy of formation can be used to bias systems toward thermodynamic heteroleptic assembly. More subtly, we have also seen how fine control of the balance of kinetics in a system can enable the formation of kinetically trapped heteroleptic products without preventing the error checking that is vital to the self-assembly of complex architectures.

LOWER-SYMMETRY LIGANDS: USING
REDUCED-SYMMETRY LIGANDS LEADS TO REDUCED-SYMMETRY PRODUCTS The complexity of metal−organic architectures may be increased through the use of components that themselves have more complex structures. This concept has recently been reviewed by Lewis and Crowley. 130 Reduced-symmetry ligands can also lead to an increased number of possible structures. Thus, we also evaluate factors that drive the selective formation of one structure from among multiple possibilities.  3.1. Reduced-Symmetry Ligands M 2 L 4 cages using bis-monodentate ligands and square-planar metal centers have been well-studied and would not be considered "complex" in terms of the scope of this review. 131,132 However, several recent publications have reported the formation of M 2 L 4 structures with reduced-symmetry ditopic ligands and a single type of metal ion 133−137 or two different types of metal ion, 138 leading to greater structural complexity. When M 2 L 4 structures assemble from a reduced-symmetry ditopic ligand, several isomers are possible ( Figure 18). Often one or more of these isomers are of lower energy than the others and therefore form preferentially.
Lewis and co-workers showed that the identity of the preferred isomer of a Pd II 2 L 4 cage can be controlled by changing the identity of the ligand (76−79). Hindered ligand 76 produces a C 2v -symmetric trans-Pd II 2 L 4 isomer in MeCN, minimizing steric clashes, with product identification being supported by DFT calculations. 133 Upon an increase in the polarity of the solvent by the use of DMSO, a mixture of the trans-Pd II 2 L 4 and cis-Pd II 2 L 4 isomers form. This phenomenon is tentatively attributed to selective stabilization of the cis-Pd II 2 L 4 isomer by the more polar solvent, which is predicted by DFT to have a larger dipole moment than the trans isomer.
The C 2h -symmetric cis-Pd 2 77 4 isomer forms selectively in DMSO. 133 This selectivity arises from the presence of different binding sites at the two ends of ligand 77, a pyridine and an isoquinoline. Within 77, the planes orthogonal to the coordinate vectors of the nitrogen donor atoms no longer coincide (even when the pyridine and isoquinoline rings are coplanar), thus favoring cis-Pd II 2 77 4 formation. Subsequent investigations involving ligand 78 indicated that in this case the deviation from coplanarity was not significant enough to yield a single isomer of the Pd II 2 78 4 complex. However, the greater steric hindrance around the coordination sphere of Pd II bound to 79 results in the formation of a single Pd II 2 79 4 isomer. On the basis of DFT calculations, cis stereochemistry was inferred.
Finally, the addition of steric bulk, in this case via the inclusion of methyl groups in 76 or 79, causes an increase in the helical twist of the structure compared with analogous structures formed by ligands lacking methyl groups. The steric effects of these methyl groups on the conformation of the resulting structure may enable tailoring of the internal cavity space.
The Lewis group has also shown that reduced-symmetry ditopic ligands containing 1,2,3-triazole and isoquinoline binding sites can form a similar Pd II 2 L 4 cage as a single cis-Pd II 2 L 4 isomer. 137 Variation of the substituent on the triazole moiety results in the formation of a series of externally functionalized cages. Because of the uniformity of the main ligand framework among all of the derivatized ligands, dynamic libraries of mixed-ligand cages are obtained when mixtures of the different ligands are used.
Bloch et al. recently demonstrated the use of conformational flexibility in producing reduced-symmetry ligands. 139 In their system, a dicarboxylate ligand with a diimine core exists in three different rotational conformations, one of which has lower symmetry. Depending on the crystallization conditions, three distinct cage isomers are isolated from a dynamic library; their structures were determined by single-crystal X-ray crystallography. The three cage isomers each contained either two or four ligands in the reduced-symmetry conformation.
Separate studies reported by Ogata and Yuasa 134 and Crowley et al. 138 also involved the formation of M 2 L 4 structures with unsymmetrical ditopic ligands. Both utilized the differing labilities of coordination bonds involving different monodentate donors or metal ions to develop mechanisms for guest capture and release. Yuasa et al. altered the stoichiometric ratio of ligand to metal in the reaction mixture to drive the interconversion of a Pd II 2 80 4 cage, capable of binding anions within its cavity, and a Pd II 80 4 complex, which does not bind guests ( Figure 19). In this mononuclear complex, the imidazole groups of all four ligands are bound to the Pd II center and the four pyridyl donors remain free because imidazole is a stronger donor than pyridine. 134 An approach introduced by Crowley et al. is based on the design and synthesis of a cage in which the antipodes are Pt II , which forms more inert Pt II −pyridyl bonds, and Pd II , which This stimulus-induced opening and closing of cage 83 also brings about reversible guest uptake and release, illustrating a potential function. Although these structures are relatively simple, they exemplify how functionality can be introduced by the use of reduced-symmetry ligands. These principles may be combined with other rules, detailed elsewhere in this review, that guide the formation of larger and more complex structures to yield architectures of greater complexity and functionality.
Hooley and co-workers reported the use of a prochiral ligand in the assembly of a desymmetrized Fe II 4 L 6 architecture ( Figure  21). 140 The presence of a prochiral CHOH center in the fluorenone liganda motif that they have explored to generate functional capsules 141−144 brought about the selective formation of "wizard's hat" 85, a distorted tetrahedron. The formation of this unusual architecture is favored by a specific pattern of hydrogen bonding involving the −OH groups at the prochiral carbon atoms of the ligands and a templating perchlorate ion at the base of the assembly. An interesting aspect of this assembly is the presence of three mer Fe II centers at the base of the structure, which are rare in self-assembled pyridylimine architectures and often drive the assembly of more complex structures, as discussed in subsequent sections.
Along with reduced-symmetry ditopic ligands, tritopic ligands with reduced symmetry can generate complex metal−organic architectures. Su et al. demonstrated the use of such tritopic ligands to form unusual architectures in the preparation of a Ag I 6 L 6 tubular structure using an elongated T-shaped ligand. 145 Hu et al. used 5-(pyridin-4-yl)isophthalic acid (87) with ptert-butylthiacalix [4]arene (86) and Ni II Cl 2 to form Ni II 40 coordination cage 88, with a structure corresponding to the J 17 Johnson solid. 146 As illustrated in Figure 22, the structure of 88, a gyroelongated square bipyramid, consists of 10 Ni 4 -p-tertbutylthiacalix [4]arene shuttlecock-like vertices and 16 panels of ligand 87. Four ligands converge at two of the 10 vertices, and five ligands converge at each of the other eight, closing the faces of the structure. In order to form the structure, the ligands coordinate to Ni II centers through different donor atoms: through the carboxylate, which can either bridge or chelate Ni II , and through the nitrogen donor of pyridine. The phenoxo oxygen and sulfur atoms of the p-tert-butylthiacalix [4]arene units also coordinate to Ni II , along with additional 87 units that do not cap the faces of the structure, DMF molecules, chloride ions, and degradation products of DMF in order to satisfy the coordination geometry of Ni II .
Hong et al. employed tritopic ligand 89, which has three binding sites arrayed asymmetrically along its length ( Figure  23). The combination of this reduced-symmetry ligand, Ni II (ClO 4 ) 2 , and pyrazole (Pz) in ethanol yields Ni II 9 89 6 Pz 6 barrel structure 90. 147 In 90, the pyrazole plays two roles, acting as a Lewis base and as an additional donor to satisfy the octahedral coordination sphere of Ni II .
Li et al. explored the use of desymmetrized tetratopic ligands resembling trapezoids to form metallosupramolecular architectures. Upon combination of these ligands with 180°d ipalladium(II) acceptors, ring-in-ring 148 or 2D Star-of-David 149 structures form.
The reaction of these same ligands with "naked" palladium-(II) ions yields three-dimensional structures. One example is Pd II 24 91 24 sphere-in-sphere architecture 92 ( Figure 24), which forms from ligand 91 and Pd II . 148 The authors drew a contrast between their approach and the one pioneered by Fujita and coworkers. 150 The Fujita approach is based on the orthogonal   assembly of two ditopic units into "independent" M 12 L 24 spheres connected via flexible linkers to give the M 24 L 24 sphere-insphere. In Li's system ( Figure 24) precise preorganization of the entire 3D architecture is enforced by the rigid nature of the ligand. Ligand 91 also reacts with a tritopic platinum(II) unit to form a double-layered pentagonal prism. 151 Ligand 91 has donor groups arrayed in two distinct ways; Li et al. also designed ligands with four distinct binding sites that form double-layered macrocyclic structures. 152 Their reports exemplify how rational design of new classes of ligands can allow unique metallosupramolecules with high degrees of complexity to be formed.

Additional Donor Sites
Another approach to designing ligands capable of forming architectures with greater complexity is the modification of ligands that have previously been used to form metal−organic assemblies, for example by appending additional donor sites. This approach was used to design pentatopic ligands 93 and 94 ( Figure 25), which form 3D hexagonal-prismatic structures 95 and 96, consisting of two connected 2D double-rimmed "Kandinsky circles", when combined with octahedrally coordinated cadmium(II) ions. 153 Ligands 93 and 94 are based upon a tetratopic donor previously reported by Li et al., 154 with the fifth terpyridine group appended to allow the two circles to be linked.
As well as providing a method for the formation of 3D structures from known 2D structures, the ligand-modification approach can be used to increase the complexity of an existing 3D structure. Fujita and co-workers employed this approach to form a Pd II 18 97 24 stellated cuboctahedron 99 using tripyridyl ligand 97, consisting of a rigid bipyridyl unit with a third pyridyl moiety flexibly tethered to the backbone. 155 As shown in Figure  26, the assembly process occurs in a stepwise fashion. The tripyridyl ligand combines with Pd II (BF 4 ) 2 to yield Pd II 12 97 24 cuboctahedron 98, analogous to a previously reported complex that incorporates a rigid bipyridine ligand. 156 Initial selective complexation with just one type of pyridine donor to form 98 is perhaps surprising at first sight. The authors suggested that the selectivity observed is due to the high kinetic stability of the cuboctahedral framework. Previous work had shown that ligand exchange on a completed cuboctahedron occurs with a half-life of 20 days. 157 Kinetic trapping of the cuboctahedron thus drives the selective assembly.
Subsequent addition of more Pd II (BF 4 ) 2 to intermediate structure 98 resulted in capping of the square faces by the coordination of four "free" pyridyl groups to each new palladium(II) center and consequent stellation of the structure to form 99. Stellation is reversed by the addition of N,N,N′,N′-TMEDA, resulting in the reformation of 98. The authors noted that this reversible opening and closing through stellation may have future applications in guest capture and release.

Nonplanar Macrocyclic Ligands
As shown in the system in Figure 22, macrocycle-derived subunits can be employed to construct coordination cages. 146,158−161 These components often have greater complexity than simpler small-molecule ligands, while still maintaining high symmetry, which increases the complexity of the resulting metal−organic architectures. 162,163 Furthermore, the use of macrocycle-derived components also may enable combination of the guest-binding abilities of the macrocycles with those of the higher-order superstructures that the macrocycles form. 164−167 Complementing the work of Hu and co-workers, who used ptert-butylthiacalix [4]arene to form the vertices of a metal− organic polyhedron, 146 macrocyclic components have also been employed as the edges and faces of metal−organic cages. Hardie and co-workers reported foundational work in this area using tritopic cyclotriveratrylene (CTV)-related ligands. 168    The Hardie group's use of CTV-related ligands to provide an array of new structure types culminated in the report of a "Solomon's cube", 170 based upon the topology of a Solomon link. 171,172 The combination of extended tris(pyridyl)cyclotriguaiacylene (100) with Pd II (NO 3 ) 2 in DMSO results in Pd II 4 100 4 structure 101 shown in Figure 27. While resembling a Solomon link, 171,172 with alternating under and over crossing points of two rings, 101 has additional connections between the rings, linking them. Consequently, the structure was described as a "Solomon's cube", with square faces and eight triply connected vertices. Structure 101 may thus be considered in terms of its three stereochemically distinct subunits: ligand 100, the Solomon link, and the figure-eight motifs lying on each of four sides of the structure. The crystal structure shows two enantiomers, in which all three of these elements concertedly show opposite handedness.
The driving force for the formation of smaller Pd II 4 100 4 assembly 101, as opposed to a Pd II 6 100 8 structure, is likely due to the fact that ligand 100 contains m-pyridine-based arms, as opposed to a linear para ligand regiochemistry, connected to a rigid macrocyclic core. Further stabilization of this topology may come from interligand π-stacking interactions.
Structure 101 in Figure 27 thus demonstrates the ability of nonplanar macrocycle-based ligands to produce more complex structure types than would be observed in analogous cases using planar D 3h -symmetric ligands. Interwoven 101 also exemplifies how the use of novel classes of ligands can lead to serendipitous discoveries.

Metallosupramolecular Chemistry Meets DNA Nanotechnology
Many of the architectures discussed in this review are assembled using small-molecule organic ligands and metal ions. A more exotic example is provided by the metal−nucleic acid cages of Sleiman et al. 173 (Figure 28). These structures require stepwise assembly of oligonucleotide strands (102). First, triangles 103 with corners consisting of two 2,9-diphenyl-1,10-phenanthroline ligands (dpp−dpp) are formed through hybridization of three complementary oligonucleotide strands. Second, two triangles are linked with single strands to give 104, and the struts are then rigidified to form trigonal-prismatic structures 105. Finally, site-specific metalation involving the coordination of Cu I , Ag I , Au I , Zn II , Co II , Cd II , or Eu II to the dpp−dpp sites, enables the creation of metal−DNA cages 106. 173 The Sleiman group also demonstrated that the order of the steps could be swapped: premetalation of the triangles followed by single-strand triangle linkage and rigidification results in the same metalated trigonal-prismatic structures. Although the flexibility in the order of construction steps indicates that metal−ligand coordination is not required to template the formation of these trigonal-prismatic structures, metalation of the structures increased their resistance to both chemical and thermal denaturation compared with their demetalated counterparts. Metal coordination was thus demonstrated to enable the formation of robust architectures assembled from strands of DNA, potentially enhancing the range of applications of 3D DNA architectures. 174−178 Through highlighting some key examples of complex or reduced-symmetry ligands that have led to novel structures, this section has emphasized the roles of both rational design and serendipity. As a general approach, the use of reduced-symmetry and complex ligands often involves rational design, sometimes with the aid of computational predictions. Postassembly rationalization has in many cases also played a role, enabling the discovery of new assembly rules, which may then be used for future designs.

LIGAND FLEXIBILITY DRIVES STRUCTURAL COMPLEXITY
Flexible ligands in many cases assemble into high-symmetry architectures. 179−185 However, flexibility within a ligand can also extend the scope of structure types beyond those having high symmetries. This section summarizes novel structure types generated via the incorporation of flexibility into the building blocks used to assemble discrete structures. Ligand flexibility often generates serendipitous results, as ligand degrees of freedom are deployed in unforeseen ways.

Flexible Ditopic Ligands
Ward and co-workers pioneered the construction of metal− organic architectures with flexible ditopic ligands, focusing on ligands containing two bidentate pyrazolylpyridine chelating sites, each attached to a central aromatic group via flexible methylene linkages. These ligands were combined in a 3:2 ratio with octahedral metal centers to yield several distinct structure types. Some of these structures have the geometries of Platonic solids, such as tetrahedra, 179−181 and others have lower symmetries and greater complexity. Several of Ward's M 8 L 12 structures exhibit symmetry reduced from that of a cube. 186,187 For example, as shown in Figure 29a, the combination of 107 with Zn II yields Zn II 8 107 12 cuboid 108 with S 6 symmetry. An antipodal pair of Zn II centers define the S 6 axis of the structure. These metal centers have fac stereochemistry but opposite handedness. 186 The other six metal centers have mer stereochemistry and are grouped into two sets of three. All of the metal centers within the same set have the same handedness, opposite to that of the other set. Mass spectrometry data showed the formation of an analogous M II 8 L 12 structure, Co II 8 107 12 , from Co II and 107. With anthracene-cored ligand 109, structure 110 was formed, which has the same M II 8 L 12 composition as 108 but significant structural differences. Cuboid 110 consists of two connected Zn II 4 109 4 cyclic helical units (Figure 29b). 187 Within each tetrameric unit, the four metal centers are trischelated in a mer fashion and have the same absolute configuration. However, as shown in Figure 29b, the handedness of the four metal centers in one tetrameric unit is opposite to that of the metal centers making up the other tetrameric face. The use of Cu II (BF 4 ) 2 with 109 yields Cu II 8 109 12 , which has a similar structure as 110. The diversity of structures formed using such ligands was further demonstrated by the formation of unusual Ni II 4 L 6 "square" and M II 6 L 9 (M II = Zn II , Co II ) "open book" structures using 107 and its modified derivatives. 188,189 Ligand 111 reacts with Ni II (BF 4 ) 2 in a 3:2 ratio in MeOH/ CH 2 Cl 2 ( Figure 30) to yield a Ni II 8 111 12 structure, which was initially thought to be cubic. 190 However, X-ray crystallography showed that product 112 has an unusual structure, based on C 2vsymmetric cuneane formed by the rearrangement of two edges of a cube (Figure 30b). All eight of its metal centers have mer stereochemistry. Interestingly, seven of the metal centers have the same absolute configuration, with the eighth displaying the opposite handedness.
Each of the two triangular faces of 112 is made up of an M II 3 111 3 metallomacrocycle (Figure 30c). Such M 3 L 3 units have been observed in structures employing similar ditopic ligands. 191−193 The four structure types shown in Figure 31 are built from M 3 L 3 subunits, with their different geometries arising from differences in how these subunits are connected to each other.
As shown in Figure 31a, structure 114, a Cd II 16   Chemical Reviews pubs.acs.org/CR Review (Figure 31b). The fac-configured Cd II centers of the Cd II 113 3 units (red spheres in Figure 31a,b) cap each of the four hexagonal faces of a Cd II 12 distorted truncated tetrahedral core described by the 12 mer-configured centers (beige spheres in Figure 31a,b) of the four Cd II 3 113 3 units. When ligand 113 reacts with Cu II , the smaller Cu II 6 113 9 trigonal-prismatic structure 116 forms (Figure 31c). Trigonal prism 116 consists of two Cu II 3 113 3 circular helical units bridged by three ligands, with some offset between triangular faces leading to distortion toward a trigonal-antiprismatic structure (Figure 31d).
The reaction of Ni II (BF 4 ) 2 with 113 produces a Ni II 8 113 12 cubic cage, which does not contain trinuclear helicate units. The observation of different structures with the same ligand but different metal ions was attributed to variations in the ionic radii and stereoelectronic preferences of the metal centers. 191 Furthermore, reaction of the same ligand (113)  The fourth structure type, shown in Figure 31g, is an M II 12 121 18 truncated tetrahedral cage framework with idealized T symmetry. This structure results from the reaction of 121, which has a naphthyl central linking group, with Cu II , Co II , or Cd II . 193,194 These structures consist of four M 3 121 3 circular helical motifs that are connected directly by six bridging ligands.
A common thread linking the different geometries shown in Figure 31 is the presence of linked M II 3 L 3 circular helicate subunits, where the three metal centers have a mer trischelate geometry. Another important feature of these four structure types is the prevalence of interligand aromatic stacking interactions, often between electron-rich central aromatic moieties on one ligand and electron-deficient pyrazolylpyridine units on another. 191−193 This elegant work by the Ward group has thus established the utility of relatively simple, flexible ligands in the construction of assemblies with structures beyond the Platonic solids, whose geometries are controlled by subtle variations in reaction conditions and ligand structure. 1 H NMR spectroscopy and mass spectrometry showed that the Cd II 16 113 24 structure 114 described above is initially present in solution, but the structure rearranges to give a smaller Cd II 6 113 9 trigonal prism over weeks in solution. 191 Replacing the 1,4-phenyl moiety in 113 with the 1,4-naphthyl in ligand 111 results in a Cd II 16 111 24 tetracapped truncated tetrahedron (in contrast to the cuneane structure observed for 111 with Ni II , shown in Figure 30a), which does not rearrange in solution. The additional interligand π stacking provided by the naphthyl spacer was inferred to stabilize the tetracapped truncated tetrahedron in solution. 195 In contrast, the reaction of 111 with Cu II does not selectively yield any species analogous to those shown in Figures 30 and 31. Instead, crystals of an unusual Cu II 12 111 15 structure form in low yield, consisting of two Cu II 3 111 3 units linked by an equatorial belt of six Cu II ions, each with a coordination number of 4 or 5. 195 Utilizing ligand 111 also allowed Ward et al. to analyze the Cd II 111 12 117 4 analogue of the structures shown in Figure 31e,f in solution. Cd II 111 12 117 4 was shown to exist as three different diastereomers in solution, with T, C 3 , or S 4 symmetry. 196 The difference between the diastereomers arises from the different relative helical handednesses of the four Cd     199 Mirkin and co-workers developed the "weak-link approach" to forming reduced-symmetry structures with complex functions. 200 Figure 33 shows a dimeric capsule produced using this approach, incorporating resorcin [4]arene and calix [4]arene subunits linked by platinum(II) centers. 201 In the absence of chloride, "weak-link" thioethers coordinate to platinum(II) binding sites. Upon the addition of chloride ions, these thioethers are selectively displaced, causing expansion of the cavity. The addition of silver(I) tetrafluoroborate reverses this expansion by abstracting chloride from the platinum(II) centers and regenerating the closed state of the capsule. In the thioethercoordinated form 131, estradiol 133 is bound selectively. In the chloride-coordinated form 130, two molecules of dextro-methorphan·HCl (132) bind instead. Sequential addition of chloride to 131 and silver(I) tetrafluoroborate to 130 brings about reversible binding and release of dextromethorphan, showcasing the ability to reversibly generate cavities with different sizes and shapes and thus control guest binding.

Flexible Tritopic Ligands
The combination of flexible tris-formylpyridine subcomponent 134 with Cd II (OTf) 2 and p-toluidine (135) yields a mixture of three products (Figure 34). 202 Two of these are T-symmetric Cd II 4 L 4 tetrahedra (137 and 138). In 137, the central methyl groups of the ligands point inside the cavity (endo), whereas in 138 these methyl groups point outward (exo). The third, minor, product is Cd II 8 L 8 tetragonal antiprism 139 with D 4 point symmetry. The eight metal centers defining the vertices of the structure have the same handedness, each with a mer arrangement of ligands.
The relative amount of the Cd II 8 L 8 antiprismatic structure 139 grows with increasing concentration because of a reduction in the entropic penalty of forming a larger Cd II 8 L 8 species instead of the smaller Cd II 4 L 4 complexes. Even more effective at driving the formation of the Cd II 8 L 8 structure is the use of 2-(4aminophenyl)ethanol (136) as a subcomponent in place of 135 and the use of a 1:3 CH 2 Cl 2 /MeCN solvent mixture. We hypothesized that these conditions allow the formation of stabilizing hydrogen-bonding interactions between the hydroxy groups of the aniline residues in the Cd II 8 L 8 antiprismatic structure. In this example, the analysis of a serendipitous result enabled the rational development of design principles for the optimized preparation of a complex architecture, illustrating the synergy between serendipity and rational design.
Hong et al. used a tris(pyridine) ligand, which had a flexible core similar to that of 134, for the construction of open Ag I 6 L 4 cages upon reaction with Ag I BF 4 . 203 These cages undergo further assembly to produce higher-order polycatenanes and polycages, depending on the reaction conditions.     (Figure 35b,c). Instead the Ce 8 L 6 complexes 143 and 145, respectively, are formed. Assembly 143 consists of a Ce 6 142 3 trigonal-prismatic framework, with two additional metal centers and three ligands forming a helical pillar within the prism. Two of the tridentate moieties of each ligand in the helical pillar bind to the apical cerium centers, and the other two tridentate sites chelate two of the metal centers making up the prismatic framework ( Figure  35b). 206 In contrast, 145 has a cagelike structure in which the flexible ligand twists so that the four cerium centers binding to the same ligand are not coplanar. 207 Furthermore, stacking interactions between the benzyl groups on neighboring ligands are inferred to stabilize the unusual structure of 145. 207 Sun and co-workers also reported the use of flexible tetratopic ligands in the synthesis of unusual "conjoined twin-cages". 208,209 They were further able to control which species formed, either a Pd II 12 L 6 cage with three mechanically coupled cavities or two helically isomeric Pd II 6 L 3 cages, by the judicious choice of assembly conditions. 208

Flexible Ligands Containing More than One Type of Coordinating Motif
This section considers flexible ligands that bind metal centers using more than one type of donor atom or binding moiety incorporated into the same ligand. Octanuclear helicate 147 (Figure 36), with a cavity large enough to bind amino acids enantioselectively, exemplifies this approach. 210 The combination of Zn II Cl 2 and chiral salen-based ligand 146 produces 147, which consists of two bowl-like Zn II 2 146 2 dimers linked by four equatorial zinc centers. Within each dimer, each five-coordinate zinc center is chelated by the N 2 O 2 pocket of one of the ligands, and the two metal centers are linked by two phenalato oxygen atoms. The two pendent pyridyl groups of each ligand remain free to coordinate to additional Zn II ions, whose tetrahedral geometries are satisfied by coordination of two chloride ions, resulting in the formation of the Zn II 8 (125), and Co II . 211 The 12 ligands form in situ and combine with 12 OH − ions, four water molecules, four octahedral Co III centers, four tetrahedral Co II ions, and 12 distorted square-pyramidal Co II centers to form a Tsymmetric tetartoid structure (Figure 37a). Furthermore, the addition of D-or L-menthol during self-assembly yields enantiopure ΔΔΔΔ-152 or ΛΛΛΛ-152, respectively. The imidazolyl 2-methyl substituent was an effective steric structuredirecting feature. This methyl group points inside the pentagonal face of the structure, whereas it could not fit within the smaller window of a cube.
In contrast, when 5-methyl-4-formylimidazole (150) or 4formylimidazole (151) combines with Co II and m-xylylenediamine, cubic cage 154 or 155 (Figure 37b) forms; in 154, the 5methyl groups point away from the faces of the structure. 212 Thus, the substituent at the 5-position of the imidazolyl ring does not exert steric control over the structure formed, in contrast with the 2-substituent. This work, together with Kwong's ( Figure 32), 197 highlights the role that flexible subcomponents play in directing self-assembly. The same simple diamine subcomponent formed complexes with very different structures depending on the steric properties of other subcomponents within the system.
Li recently reported the use of a different flexible bis-(imidazole) ligand, 156 (Figure 38), to form bicapped squareantiprismatic structure 157 upon reaction with Cu II under solvothermal conditions (Figure 38). 213 Single-crystal X-ray diffraction showed the formation of Cu II 10 156 8 cages that have two types of Cu II centers. Eight equatorial Cu II ions have a distorted square-pyramidal geometry, with tetradentate chelation of one ligand and monodentate binding of a second. Each of the two axial Cu II centers is bound by four imidazolate donors, with additional coordination of anions and water molecules to complete the coordination sphere.
Bicapped square-antiprismatic structure 157 can expand or compress vertically to accommodate different anions in its cavity because of its flexible ligands. Among the anions encapsulated (SiF 6 2− , ClO 4 − , Br − , and Cl − ), SiF 6 2− gives the largest cavity volume and Cl − the smallest. Cage compression is triggered through anion exchange, for example, by the addition of KCl to a cage binding ClO 4 − internally. Li et al. also employed ligand 156 to form a mixed-valence Cu II /Cu I metallocycle. 214 Upon combination of this metallocycle with triethylenediamine in a 2:3 ratio, a trigonal-prismatic structure forms. 215 This trigonal prism undergoes a structural transformation to form 157 upon oxidation of Cu I to Cu II .

Ligand Flexibility Arising from Substituent Positioning
An alternative way to introduce flexibility into ligands without incorporating alkyl or other flexible linkers is to vary the position of substitution of aryl rings or change the metal-binding moieties so as to provide multiple conformers capable of binding metal ions in different ways.
For example, tri-and tetratopic ligands that employ 3-pyridyl binding sites or imidazoles have been used in place of conformationally locked 4-pyridyl binding sites. When binding to cis-protected square-planar metal centers, such tritopic ligands can form M 6  and Diels−Alder reactions within its hydrophobic cavity in water. 216 Recently Klajn and co-workers adopted this cage for the investigation of photoswitching in confined environments. 218    Chemical Reviews pubs.acs.org/CR Review ligand flexibility was employed by Schroder and co-workers to form a Cd 66 nanosphere with idealized T symmetry. Its dualshell structure consists of a sphere of 66 Cd II centers bridged by μ 3 -hydroxide, μ 3 -oxo, and μ 5 -NO 3 − anions and enclosed by 12 DMF ligands and 20 tritopic organic capping ligands. 225

Flexible Pseudolinear Polypyridyl Ligands
Fujita and co-workers have demonstrated that in addition to the formation of nanotubes from relatively rigid polypyridyl ligands through guest templation (section 7.4), nanotubular structures are also obtained using more flexible ligands. The combination of Pd II (en)(NO 3 ) 2 with ligand 160 ( Figure 40) and a rodlike guest template results in the formation of Pd II 6 160 end-capped tube 162. 226 The flexible nature of the benzenetetracarboxylatecontaining core of the ligand allows it to fold and form structure 162 containing only one folded ligand. Selective guest binding within this tube was observed, whereby a biphenylcarboxylate guest bound unidirectionally with the biphenyl group ensconced in the hydrophobic pocket of the tube and the hydrophilic carboxylate exposed to the solvent.
At higher concentrations, the longer Pd II 12 160 2 tube 163 forms as a minor species and was isolated via crystallization. Xray crystallography revealed a doubly open-ended tube that is 3 nm in length with two template molecules residing inside the cavity.
Similarly, Chand et al. showed that a flexible pseudolinear tripyridine ligand forms a Pd II 3 L 4 double-decker cage. 227 Upon reduction of the metal:ligand ratio from 3:4 to 1:2, the ligand reconfigures into a U-shaped conformation in which the two terminal pyridines bind to the same Pd II center to form a Pd II L 2 spiro-type complex and the central pyridine donor of each ligand remains uncoordinated. Interconversion between the two structure types occurred following alteration of the metal:ligand ratio of the reaction mixture.
A consistent theme for this section is that the structures formed from flexible ligands can be difficult to reliably predict, meaning that the results are often serendipitous. However, as elsewhere, rules and hypotheses derived from these initial observations can enable the design of related structures and components to selectively form a desired structure that may have initially been observed to form as one component of a mixture. The adaptability exhibited by some structures formed using flexible ligands is more rarely observed for structures formed with more rigid ligands. The reconfiguration of these more flexible structures can lead to new functions, often related to guest binding.

COMPLEXITY DERIVED FROM SOLVENT, ANIONS,
AND TEMPLATES Changes in the environments where metal−organic cages form can alter the structure formed. The course of self-assembly may be reconfigured by changing the solvent, adding a guest, or manipulating external conditions such as temperature and concentration. Although the effects of the environment on the structure may be challenging to predict beforehand, they may be rationalized, and the resulting knowledge can again be used to infer self-assembly rules. In this section we review techniques used to generate complex architectures from simple subcomponents by modulation of the external conditions and by guest addition.

Solvent-and Concentration-Dependent Complexity
One of the most straightforward methods to direct the assembly of complex architectures is to vary the solvent used. We reported a system where tetrahedral metal−organic cage 165 forms in water but a mixture of methanol and water leads to the selective formation of pentagonal antiprism 166 instead (Figure 41). 228 By tuning of the temperature in addition to the solvent, either architecture can be prepared exclusively, with lower temperatures favoring the pentagonal antiprism. This process involves a switch from fac coordination stereochemistry around the metal centers in the tetrahedral cage to all mer metal centers in the pentagonal antiprism, where the lower-symmetry mer coordinative linkages give rise to increased structural complexity. 229−232 Antiprism 166 is kinetically stable toward changes in the solvent, requiring heating for a week to convert to the thermodynamically preferred tetrahedron 165 following solvent exchange. However, the pentagonal antiprism is responsive to the addition of a competing aniline: addition of 4-methoxyaniline to a mixture of pentagonal antiprism 166 and tetrahedron Chemical Reviews pubs.acs.org/CR Review 165 brings about the selective disassembly of 166. Severin and co-workers were able to trigger the rearrangement of an octanuclear prismatic cage that forms in chloroform into a tetranuclear complex by exchanging the chloroform solvent for dichloromethane. 233 This work illustrates how even quite subtle changes in the solvent can trigger substantial transformations in the assembly structure, especially when the structures have specific binding interactions with that solvent. Newkome, Wesdemiotis, and co-workers reported an assembly process where at higher concentrations ligand 167 assembles with Cd II to produce bisrhombohedral structure 168, whereas at lower concentrations the simpler tetrahedron 169 is favored (Figure 42). 234 The use of tris(terpyridine) ligands with Cd II provided the delicate balance of lability and stability required for these structures to form. At higher concentrations, the bisrhombohedral architecture forms exclusively, whereas the tetrahedron is the exclusive product at lower concentrations. To confirm the structure of the tetrahedron, which could not be isolated because of rapid equilibration back to the bisrhombohedral architecture, a ruthenium(II)-containing metalloligand, essentially consisting of two units of 167 connected by the coordination of one (blue) terpyridine unit on each 167 to a kinetically inert ruthenium(II) center, was employed. The formation of a similar tetrahedral architecture confirmed the structural assignment of 169. The authors subsequently reported a system wherein the predominant product among three architecturesa cuboctahedron, an octahedron, and a triangular sandwich complexdepended upon the concentration. 235 These systems provide a way to generate and switch between complex architectures selectively in solution through manipulation of the concentration.
Shionoya and co-workers reported that tritopic ligand 170 forms bowl-like structure 171 and pseudotetrahedron 172 ( Figure 43). 236  Shionoya's group reported the use of a similar ligand with fourfold symmetry in combination with zinc triflate and a mixed solvent system to generate an unusual D 3 -symmetric enneahedron. 237 We have also made use of solvent effects in a system where a tetrahedron interconverted with dimeric and trimeric stacked structures on the basis of different chemical stimuli. 238 Analogous stacked structures had previously been generated from a more rigid, achiral subcomponent, where interconversion between double and triple stacks was controlled by subcomponent substitution. 239

Temperature-Dependent Assembly
Hiraoka and co-workers reported the intriguing system shown in Figure 44, where two similar building blocks sort with a selectivity that varies as a function of temperature. 240 Although these capsules are purely organic ( Figure 44) and are held together by van der Waals forces, cation−π interactions, and the hydrophobic effect, their novel mechanism of sorting warrants inclusion in this review. The authors used gear-shaped amphiphilic molecules 173 and 174 with hexaphenylbenzene cores that self-assemble to form hexameric cubic architectures. These two hexaphenylbenzenes differ in the presence (173) or absence (174) of methyl groups at three positions ( Figure 44).   Chemical Reviews pubs.acs.org/CR Review These additional methyl groups have a significant effect on the thermal stability of the formed hexameric architectures, with assembly 175, composed of trimethylated 173, dissociating at 130°C, whereas assembly 181, formed from non-methylated 174, dissociates at 65°C. When 173 and 174 were mixed and the mixture was allowed to equilibrate at room temperature in water, a statistical distribution of capsules 175−181 containing both panels was formed as a result of the structural similarities between them. The authors then heated the scrambled system above the disassembly temperature of 181, which led to survival of only 175 at 100°C. The authors then quenched the system by rapid cooling, trapping it in a metastable state consisting of only the two homogeneous capsules 175 and 181. These capsules then reequilibrated, taking 2 days at 25°C to reach the equilibrium state of a statistical distribution. This process could be further controlled by binding of guests in the cavities of these self-assembled architectures. This work provides a unique example of control over the statistical distribution of a system of capsules using temperature-quenching techniques analogous to those used in metallurgy.

Guest-Templated Assembly
Fujita and co-workers prepared self-assembled heteroleptic trigonal prism 184 (Figure 45), the assembly of which required the presence of certain π-extended guest molecules. 241 When no guest was present, oligomeric species and homoleptic octahedral [(ethylenediamine)Pd II ] 6 10 4 predominated. The selective formation of heteroleptic 184 is driven by aromatic stacking interactions between electron-poor cage panels (10) and coronene (183) and also by the steric bulk of the linear bipyridine strut 182, which disfavors the coordination of two linear bipyridine ligands to a single palladium center. When the bipyridine struts are extended, allowing the formation of larger trigonal prisms, three guest molecules stack selectively within the capsule cavity. Both capsules, containing two or three guests, were shown by UV/vis spectroscopy to exhibit charge-transfer character.
The Fujita group has also reported systems of capsules formed from Pd II with cis-chelating bidentate ligands and pyridine or pyrimidine donor ligands, where guest binding triggers structural transformations. In one example, a trigonal-bipyramidal cage transforms into an octahedral architecture upon guest binding. 242 In another, an octahedral Pd II 20 L 8 structure transforms to an open Pd II 8 L 4 bowl-shaped structure. 243 Further to this, Hiraoka and Fujita were able to control the specific constitutional isomer of a Pd 3 L 2 complex that forms from a reduced-symmetry tripyridyl ligand by judicious choice of guest molecules. 244 Addition of a flat guest molecule (1,3,5-benzenetricarboxylic acid) favored the formation of a capsule with a relatively flat cavity, whereas the addition of a spherical guest (CBrCl 3 ) resulted in the formation of an isomer with a more spherical cavity, providing an early example of guest control of isomer formation. We reported the transformation of a tetrahedral Fe II 4 L 6 cage with porphyrin panel edges upon the addition of C 60 or C 70 . After addition of the fullerene to the system, a Fe II 3 L 4 conelike architecture formed. Selective transmetalation of a single iron(II) vertex for copper(I) was favored, resulting in the formation of a heterometallic Cu I Fe II 2 L 4 structure, exploiting the coordinative unsaturation of iron(II) at a single position. 245 Our group also reported the formation of a cuboctahedron that shows cooperative binding of a pair of C 60 molecules. The binding of these guests triggers a rearrangement of the original O-symmetric structure to an S 6 -symmetric analogue that optimizes fullerene binding. 246 Muller and Moller reported the formation of a trigonalbipyramidal capsule that incorporated its template. 247 Sodium 5,5-diethylbarbiturate was added to a threefold-symmetric guanidinium-based ligand that chelated three Pd II centers, leading to the formation of a trigonal-bipyramidal architecture containing 33 distinct building blocks. While investigating the endohedral functionalization of metal−organic polyhedra via coordination of organophosphonates to polyoxovanadate units at the vertices of the cages, Fang et al. observed the formation of a barrel-like structure instead of a tetrahedral structure that would ordinarily be expected to form. 248 The observed preference for the barrel-like structure was attributed to steric effects, as the interior cavity of the tetrahedron would be too small to accommodate the sterically bulky organophosphonate groups. Donnelly, Abrahams, Paterson, and co-workers likewise reported the guest-induced formation of coordination nanotubes that can bind small molecules such as CO 2 , CS 2 , and acetonitrile. 249 The Yoshizawa group reported a modification of their anthracene-paneled M 2 L 4 lantern architectures wherein guest binding drives the formation of heteroleptic structures ( Figure  46). 250 They used two pyridine-based bidentate ligands of different lengths, 185 and 186. Each ligand independently assembled with the metal ion to form an M 2 L 4 lantern architecture with spherical internal cavities capable of binding aromatic guests. When the two cages were mixed together, a complex mixture of hetero-and homoleptic architectures formed. However, when C 60 was added to this mixture, a single heteroleptic host−guest species was seen, with the composition Pd II 2 185 2 186 2 ⊂(C 60 ). Computational models suggested that the cis isomer of this architecture is favored over the trans isomer.  The observed preference for the heteroleptic architecture was attributed to the optimization of aromatic stacking interactions between the guest and host. This guest-driven host rearrangement concept might be employed to selectively generate a range of architectures tailored to specific guest-binding tasks. In this approach, the cavity of the host is already filled, thus precluding its use for the binding of guests with lower affinity than the template or necessitating the optimization of template removal. Yoshizawa and co-workers recently reported the use of a guest template to drive a system of equilibrating atropisomeric cages toward a single isomer in a similar system. 251

Anion-Templated Assembly
We reported a system of complex architectures based on anion binding and subcomponent self-assembly ( Figure 47). 252 Diformylbipyridine subcomponent 187 combines with ptoluidine (135) and cobalt(II) triflimide to generate an initial mixture of architectures. The addition of a triflate template leads to the assembly of a Co II 4 L 6 tetrahedron. However, when lithium perchlorate or potassium hexafluorophosphate is added to either the dynamic library or the tetrahedral architecture, clean conversion to pentagonal prism 188 ( Figure 47) is observed upon heating.
Co II 10 L 15 architecture 188 consists of two parallel Co II 5 L 5 pentagonal rings connected by five bridging ligands, creating a barrel-like structure. This structure is templated by five perchlorate or hexafluorophosphate anions in binding pockets between pairs of bridging ligands. Forming a pentagonal prism in place of the initial tetrahedral structure is favored for reasons that include the maximization of stacking interactions and a better size match of perchlorate or hexafluorophosphate with the smaller cavities found in the pentagonal prism.
The formation of pentagonal prism 188 also generates a central binding pocket surrounded by 10 internally facing pyridine CH groups. This pocket is occupied by a chloride ion, scavenged during synthesis, which is so strongly bound that it cannot be removed by addition of silver(I), in analogous fashion to other self-assembled systems found to bind strongly to halides. 253−255 This system thus provides an unusual way to generate a secondary anion binding site by the addition of an initial anionic stimulus. Further work explored different architectures formed by subcomponent self-assembly using this same dialdehyde 187. 256−259 Chifotides and Dunbar reported the use of anion−π interactions to control the formation of tetrameric or pentameric helicates, depending on the identity of the anion used. 260 The choice of anion during self-assembly likewise dictates the identity of the product formed in the work of Pan, Xu, and coworkers, where the formation of either a Pd II 2 L 4 or a Pd II 3 L 6 capsule is driven by the addition of nitrate (favoring Pd II 2 L 4 ) or triflate/tetrafluoroborate (favoring Pd II 3 L 6 ). 261 This selectivity is driven by differential guest binding within each capsule. Luẗzen and co-workers also reported a chiral Pd II 4 L 8 flexible architecture whose formation is dependent on templation by tetrafluoroborate. 262

MULTIMETALLICS: HETEROMETALLIC AND
CLUSTER-CONTAINING ARCHITECTURES Complexity may be enhanced through the development of heterometallic self-assembled systems, which employ the differing coordination preferences of more than one metal. The development of systems that take advantage of the coordinational flexibility and unusual geometries of metal clusters can also lead to the formation of novel architectures. Either the kinetics or the thermodynamics of self-assembly may be employed to direct the outcome of a multistep process, as described in the examples below.

Ligand Coordination Preference
An early example of heterometallic supramolecular assemblies was provided by Raymond and Wong, who used ligand 189 that contains both hard and soft donors ( Figure 48) to selectively bind two different metal ions. 263,264 The catechol group of ligand 189 binds to hard metal centers such as Ti IV and Sn IV , and the phosphine binds to softer metal centers such as Pd II . A stepwise process was initially employed, first installing the catechol-binding metal, as insoluble polymers were observed when phosphine coordination was attempted first. However, under the optimized conditions Ti IV -and Sn IV -containing mesocates are generated in a single step via selective selfassembly. Similar principles using other ligand designs have been used by the groups of Wang, 265 Duan, 266,267 Brechin, 268 and Youngs 269 to generate other trigonal-bipyramidal assemblies.
Expanding upon the principles developed by Raymond and Wong, 263,264 Luẗzen et al. reported a system incorporating both Fe II , which is selectively bound by pyridylimines, and Pd II , which binds selectively to monotopic pyridine donors, to form a system capable of complex-to-complex switching with a concomitant spin-state transition ( Figure 49). 270 Selective binding is driven both by intrinsic ligand preference (avoidance of steric clash at palladium centers ligated by pyridylimines) and by the preassembly of Fe II metalloligand 193 using chelating tris(2aminoethyl)amine (192). This chelating ligand enforces fac hexadentate coordination on the assembly. Subsequent addition    4 ]-(BF 4 ) 2 instead generates cubic architecture 195, which has a structure analogous to one that we prepared using a preassembled platinum complex metalloligand. 271 When a less sterically hindered aldehyde subcomponent is added, the more hindered subcomponent 191 is displaced, and the iron(II) transitions from a high-spin state to a low-spin state.
Crowley and co-workers reported a nonanuclear heterometallic Pd II 3 Pt II 6 donut-shaped cage, where the use of two different metal-binding sites on each ligandbidentate triazolylpyridine versus monodentate pyridineenables heterometallic assembly. 272 The architecture catalyzes the lightmediated hetero-Diels−Alder reaction of anthracene with singlet oxygen.

Combining Kinetically Inert and Labile Metal Ions
The use of a mixture of kinetically inert and labile coordination centers has been explored extensively by the Ward group ( Figure  50). 273−276 Their stepwise approach involves the initial formation of a coordination complex between three pyrazolylpyridine ligands and kinetically inert Ru II or Os II . This complex is formed as a statistical mixture of fac and mer isomers, and the minor fac isomer is then isolated, exploiting the inertness of these metals. 277 An example of this fac complex, fac-Ru113 3 , is represented in red in Figure 50. One coordinating site on each ligand is occupied by the Ru II , leaving the other free for coordination to a coordinationally labile metal center. Subsequent addition of labile Co II , Cd II , or Ag I then results in self-assembly with efficient error checking, as the structures undergo coordinative reconfiguration about the labile metal ion.
The Ward group further developed this system to install two types of ligands selectively into a heterometallic architecture to form Ru II 4 Cd II 12 111 12 113 12 twisted tetracapped truncated tetrahedral array 196 ( Figure 50). 276 The design of this structure builds upon the homometallic M 16 L 24 structures previously reported by the same group, as depicted in Figure  31a. 191 The Jin group also reported a series of heterometallic capsules that combine kinetically inert metals such as rhodium and iridium with kinetically labile metals such as silver and zinc. 278

Multimetallic Vertices
Dicopper "paddlewheel" vertices have been used to generate an array of molecular capsules. 279,280 In an elegant example, Schmitt and co-workers generated "capsule-within-a-capsule" 198 using dicopper paddlewheel complexes as nodes ( Figure  51). 281 Extended tri-m-benzoic acid ligand 197, once deprotonated, reacts with Cu II (NO 3 ) 2 to form 198. X-ray crystallography revealed a complex octahedron-within-cuboctahedron architecture, with the inner assembly fully linked to the outer one. The architecture, which can be made soluble by postassembly modification with alkylpyridine donors, can absorb 7-amino-4-methylcoumarin from solution.
The potential of using coordinationally flexible bimetallic clusters was recently highlighted by our group in the assembly of   Chemical Reviews pubs.acs.org/CR Review a trigonal-prismatic cage that incorporates disilver vertices ( Figure 52). 282 The nonconverging coordination vectors of 2formyl-1,8-napthyridine (200) combined with the flexible coordination sphere of silver(I) leads to the formation of disilver vertices. The geometry of tris(4-aminophenyl)amine (199) and an appropriate anionic template (Figure 52b) generate Ag I 12 L 6 trigonal prism 201. Crystallographic analysis of 201 showed that two anions are bound within its cavity, held in proximity by the surrounding metal−organic architecture. Two HSO 4 − anions bind in close proximity, stabilized by additional hydrogen-bonding interactions, as seen in the cyanostars of Flood and colleagues. 283 Linear, covalently linked dianions also serve as competent templates for the trigonal prism. Even highly oxidizing species such as peroxodisulfate, which is known to oxidize Ag I to Ag II , 284 can be used, demonstrating the power of self-assembly to alter the properties of structural subunits. 2,35,62,285 Silver clusters have also been used to generate a Ag 180 nanocage with a diameter of 2.5 nm based on silver "trigons" (three silver ions in a triangular arrangement). 286 We further extended this concept to incorporate not just bimetallic corners but also Ag I 4 and Ag I 6 clusters as integral structure-directing motifs using the ditopic subcomponent 202. 287 As shown in Figure 53b, 202 and 203 initially form mixtures of tetrahedra and three-stranded helicates (204−207). The mixture of 204 and 206 then proceeds to generate sixstranded helicates 208 and 209 in the presence of suitable anionic templates (Figure 53c,d), while analogous structures do not form from subcomponent 203. Key to the formation of these unusual structures is the judicious choice of anion. Whereas the addition of other anions to the equilibrating mixture of 204 and 206 does not effect structural transformation, addition of iodide, bromide, or sulfate triggers rearrangement to a six-stranded helicate that resembles a sheaf of wheat. Its elongated structure was confirmed by X-ray crystallography and solution NMR spectroscopy. This work provides an unusual example of the mutual stabilization between metal clusters and a self-assembled architecture, with anions playing a central role in structuring the metal cluster and therefore the superstructure thus formed. 287 Other metal clusters have been used as capsule vertices, 288 including polyoxometalate-derived caps, 289 which formed a capsule with an unusual "near-miss Johnson" geometry; tungsten−copper synthons, which enabled the formation of distorted octahedral structures; 290 tripalladium vertices ligated by tetrazole linkers; 291 and trizirconium clusters, which formed the corners of a chiral coordination cage capable of performing sequential asymmetric reactions. 292 Manganese has also been employed to generate cages related to truncated tetrahedra. 293

Organometallic Macrocyclic Tubes
Tubular assemblies may be generated by linking macrocyclic ligands with a band of metal centers, as exemplified by the work  Architecture 211 exemplifies a novel approach to metal-driven self-assembly where the metal ions define a central ring rather than vertices. Such architectures have been used to form mechanically interlocked organometallic [2]rotaxanes. 295 In a conceptually related example, Shionoya and co-workers had previously reported a Ag 3 L 2 structure in which two ligand disks were linked by a ring of three silver ions, but solid disk-shaped ligands were used rather than macrocycles. 296 Hetero-and multimetallic self-assembled structures have not been used as extensively as other techniques to generate complex architectures, but there is clearly great potential in this approach. A focus of future work will be utilizing the properties of the multiple metal ions to achieve tasks that cannot be achieved by a single ion. The use of metal clusters with coordinational flexibility as vertices likewise shows promise in the generation of complex architectures from simple ligands.

GEOMETRIC, STERIC, AND SUBTLE
NON-COVALENT EFFECTS Geometric constraints can shape the self-assembly of highsymmetry structures. Key examples from the Fujita group have shown that even slight alterations of the bend angle (θ, Figure  55), or flexibility, of dipyridyl ligands can result in the formation of polyhedra with dramatically different sizes ( Figure 55). 297,298 The selective formation of tetrahedra or cubes also depends upon the relative orientations of the coordination vectors in bisbidentate ligands. 28 Similar geometric principles have been shown to drive the formation of more complex, lower-symmetry architectures and to enable discrimination between different structures of high complexity. This section will highlight instructive examples of geometric control along with cases demonstrating the impact of subtle steric effects and noncovalent interactions on the outcome of self-assembly processes.

Bend-Angle Dependence of Ditopic Struts
Key work by Fujita and others has demonstrated that myriad metal−organic structure types with the general formula M II n L 2n are formed through the self-assembly of dipyridyl ligands having different bend angles with square-planar Pd II and Pt II cations. These products are symmetrical polyhedra (Figure 55) 302 In contrast, carrying out the reaction in CD 3 CN results in the formation of smaller Pd 3 216 6 tube 218. The selective formation of 217 requires the presence of both DMSO and nitrate, with a mixture of 217 and 218 observed when Pd II (OTf) 2 in DMSO is used. Finally, the ratio of 217 to 218 tracks the DMSO:MeCN ratio of the solvent mixture. Using similar principles, the Yoshizawa group obtained a Pd II 2 L 4 polyaromatic capsule that was structurally contracted in comparison with a previously reported capsule using a similar ligand. 303 In targeting the next-largest structure in the series of regular Pd II n L 2n assemblies shown in Figure 55, a Pd II 60 L 120 rhombicosidodecahedron, by widening the bend angle of the ditopic ligand, Fujita et al. instead formed an unexpected new architecture. Selenophene-centered ligand 219 ( Figure 57) exhibits a bend angle of 152°, a modest increase upon the angle of 149°in the thiophene-centered ligands previously reported to assemble into Pd II 24 L 48 214 and Pd II 30 L 60 215 ( Figure  55). 297−299 The assembly of 219 and Pd II in DMSO instead Chemical Reviews pubs.acs.org/CR Review yields structure 220, with a formula of Pd II 30 219 60 , the same composition as icosidodecahedral structure 215. 304 Refinement of X-ray crystallographic data through the extension of techniques originally developed for protein crystallography revealed that the surface of structure 220 is tiled by eight triangles and 24 squares. Fujita's team developed a mathematical description of this new class of structures, which they named tetravalent Goldberg polyhedra. According to their nomenclature, 220 is a tet-G(2,1) polyhedron, with the numbers in parentheses describing the relative orientations and spacings of the triangles among its squares. In 220, a given triangle is located two steps horizontally and one step vertically away from its nearest neighbor. Structure 220 is chiral, existing as a pair of enantiomers (Figure 57b). Using graph theory, the authors predicted that Pd II 48 L 96 tet-G(2,2) structures would be the next largest members of their new series. Meticulous modeling predicted that a ditopic ligand with a bend angle of 152°should favor the formation of such a Pd II 48 L 96 architecture, suggesting that the initially observed Pd II 30 219 60 structure is a kinetically trapped species.
Through optimization of the conditions of self-assembly and screening of many crystals, the Fujita group was once more able to use their novel crystallographic methods to identify a Pd II 48 219 96 structure with the geometry of a larger tet-G(2,2) polyhedron. This remarkable structure demonstrates the power of using the analysis of serendipitous results in the targeting and discovery of new structures. Such structures are among the largest synthetic assemblies known, rivaled only by those incorporating biomolecular building blocks. For example, the Heddle group have used metal-driven assembly of protein subunits to produce large assemblies with unusual structures, such as the snub cube. 305 Newkome et al. demonstrated the role that geometric constraints can play in the assembly of heteroleptic structures. They reported that mixing hexatopic ligands 221 and 222 with Cd II in a 1:3:12 ratio results in the formation of the Cd II 48 221 4 222 12 truncated tetrahedral structure 223 with a molecular weight of approximately 70 kDa (Figure 58). 306 Each of the individual bent 222 ligands acts to connect two hexagonal 221 units, four of which make up the faces of the truncated tetrahedron. The rigidity of bent 222 enables the selective formation of the desired structure, which does not form when a more flexible alkyl linker is used as the spacer between the two tris(terpyridine) units. Instead, double-decker hexagons are formed. 307   The combination of elongated tetratopic pyridyl-based ligands and cis-protected square-planar metal centers has been shown to yield a variety of trigonal, 312−314 tetragonal, 315−318 pentagonal, 319 and hexagonal 320 barrel-like structures (224− 227; Figure 59). 321,322 Intriguingly, recent reports have demonstrated that ligands of this class can also form structures with gyrobifastigium, 323−325 triangular-orthobicupola, 326,327 and square-orthobicupola 319 geometries (228−230), in some cases selectively and in others as a minor product. Thus, while the combination of elongated tetrapyridyl ligands with cisprotected square-planar metal centers can yield different structures (Figure 59), control over the thermodynamics of the system is required to ensure that a single product is formed.
Recent work by Severin and co-workers sought to uncover the design principles responsible for the assembly of several different structures, particularly taking into account geometric considerations. 319 When the authors postulated that the ligands were fully rigid, geometric analysis predicted that larger, often pentagonal, barrels would form. However, tetragonal or trigonal-prismatic barrels are observed in practice. The formation of these entropically favored smaller assemblies is thus attributed to the conformational flexibility of the ligands, which enables them to deviate from planarity, and also to flexibility arising from the potential for a slight misalignment between the coordinate vectors of the ligand and the metal− ligand bonds.
Severin's group also discovered that ligands containing bulky cores can form structures with gyrobifastigium-like geometries (228; Figure 59). 323 This geometry allows a greater distance between bulky ligand cores, thereby reducing steric clashes compared to barrel-like geometries. Further geometric analysis indicated that the gyrobifastigium structure emerges only in a certain window of ligand length-to-width ratios. Further elongation of the ligand precludes gyrobifastigium formation, favoring instead the formation of larger barrel-like assemblies, which also reduce steric clashes between ligand cores. 319 Factors beyond simple geometric considerations were also revealed to be important in determining which among the many architectures shown in Figure 59 might predominate. For example, favorable interligand non-covalent interactions can influence the preferred geometry of the structures formed. 325 Although analysis of geometric considerations does not yet provide a definitive guide for selectively obtaining each of the seven structure types shown in Figure 59, these principles may be used to guide the targeting of other structure types, using ligands with bulky metal(II) clathrochelate cores. 319 Mukherjee et al. further demonstrated the utility of this class of structures using water-soluble tetrafacial barrel 232, selectively formed via the combination of tetrapyridyl ligand 231 and cis-Pd II (en)(NO 3 ) 2 in a 1:2 ratio (Figure 60). This barrel acts as a carrier for curcumin. 315 Complexation within the cavity increases the water solubility of curcumin and also protects it from photodegradation in aqueous solution when it is exposed to either daylight or UV light. The authors attribute the photostabilizing property of the metal−organic barrel to absorption of most of the incident photons by the aromatic panels of the structure, reducing the number absorbed by the curcumin guest. Curcumin has been shown to have pharmacological activity, 328 but two factors limiting its potential use in therapeutics are its tendency to undergo photodegradation 329 and its low bioavailability arising from poor aqueous solubility, 330 both of which may be alleviated by supramolecular carrier 232. The Mukherjee group also reported a ureafunctionalized trigonal prism capable of catalyzing Diels− Alder reactions in aqueous media, further demonstrating the potential application of water-soluble open cages. 331 Utilizing the principles discussed above, along with those illustrated in Figure 3, Zhang and co-workers constructed heteroleptic tetragonal barrel-shaped metallacages, which were emissive in both solution and the solid state. 332 We reported the formation of a series of tubular M I 8 L 4 structures 235 via subcomponent self-assembly ( Figure 61) that have narrower cavities than the other structures discussed above. They are obtained from the reactions of elongated tetraanilines (233a−c), a 2-formylpyridine (16, 234a, or 234b), and Ag I or Cu I . 333,334 The tubelike hosts exist in two possible diastereomeric forms, either D 2 /D 2d -235 or D 4 -235. The equilibria between diastereomers depend upon the ligand length, the substituents, the identities of the metal ion and the counteranion, and the temperature. 334 Furthermore, the linear cavities of the D 4 -symmetric isomers of these structures bind and   Figure 62). 338 Although an X-ray crystal structure was not obtained for 238, solution studies and molecular modeling enabled its assignment as shown in Figure 62. The modeled structure has a ligand curvature of 87°, close to the value of 86°o bserved for the free ligand. An analogous structure forms by the assembly of ligand 237 with trans-Pd II Cl 2 (MeCN) 2 in DMSO. 338 An additional feature of electron-rich ligand 237 is its ability to undergo two-electron oxidation. The conformational change of the ligand core that accompanies this oxidation was exploited to drive the redox-controlled disassembly and reassembly of a M 4 L 2 coordination cage. This redox-governed process was coupled with guest binding to provide a means of guest release and recapture. 339 Ag I 12 237 6 structure 238 rearranges from the discrete cage into a three-dimensional supramolecular polymer when the ligand units are oxidized to the dicationic state. In contrast, the Pd II 12 237 6 cage remains intact upon oxidation. 338

Linear Polytopic Ligands
The development of structures that can bind more than one guest is of great interest in the context of new coordination cage applications. 57 This goal may be achieved through the design and synthesis of metallosupramolecular structures that contain multiple linked cavities with a high degree of enclosure and are therefore expected to exhibit guest binding. Crowley and coworkers expanded on design principles for the formation of simpler Pd II 2 L 4 structures to design pseudolinear polypyridyl ligands that form multicavity structures. 93 These architectures contain similar cavities linked end to end.
As shown in Figure 63, combining hexapyridyl ligand 239 and Pd II leads to the formation of triple-cavity cage 240. A higher temperature than typically needed for the formation of Pd II 2 L 4 complexes is required in order for the error-correcting disassembly of misassembled intermediate structures to occur during the formation of multicavity structures. To illustrate the potential use of such cages, the authors demonstrated the segregated binding of two distinct types of guests within the two different cavity types (terminal and central) within 240. Cisplatin is encapsulated in the terminal cavities, whereas triflate is bound in the central cavity as well as externally at each end of the structure. 93 Clever et al. formed double-and triple-cavity cages by expanding upon these principles. A topologically interpenetrated dimeric species forms from the double-cavity cage upon   addition of chloride or bromide ions. 340 Each of the five segregated interior cavities of the structure can bind chloride or bromide.
Yoshizawa et al. reported the assembly of W-shaped tripyridine ligand 241 with Pd II to form Pd II 3 241 4 double capsule 242. 341 When structure 242 is heated with C 60 , the central Pd II is ejected, resulting in the formation of Pd II 2 241 4 · (C 60 ) 2 "molecular peanut" 243 ( Figure 64), with extensive stabilization from aromatic stacking interactions between the anthracene panels of the host and the fullerene guests outweighing the energetic cost associated with a loss of coordinative saturation of the central pyridine. Capsule 242 also simultaneously binds different guests, diamantane and phenanthrene, with the preference for heterotopic encapsulation of these guests to form 242·(diamantane)(phenanthrene) 2 complex 244 attributed to cooperative changes in the volumes of the two cavities that occur upon guest binding. The same group later showed that similar peanut-shaped polyaromatic shells form under much milder conditions when the central pyridine ring is replaced by a phenyl ring, which allows the stepwise encapsulation of two C 60 molecules. 342 Chand and co-workers reported that ligands 245−247 form the conjoined cages 248−250 (Figure 65), further developing the concepts introduced above. Instead of different cavities having similar sizes and shapes, ligands were rationally designed to form architectures consisting of laterally or linearly conjoined Pd II 2 L 4 and Pd II 3 L 6 units ( Figure 65). 343 Structures 248−250, which were all confirmed by X-ray crystallography, assemble from one or more of the carefully designed ligands 245−247 and Pd II in DMSO. In the cases of 248 and 249, integrative selfsorting results in the selective formation of heteroleptic structures. The different types of cavities bind different guests. Small anions, such as NO 3 − and Cl − , bind within the smaller cavities, and their presence is required to template the formation of structures 248−250. Crystallography also showed that multiple DMSO molecules reside in the larger, trigonal, cavity.
Fujita et al. reported the use of linear polypyridyl ligands in the formation of a series of nanotubular architectures. The combination of ligand 251 (Figure 66a) with cis-protected Pd II and a template such as 4,4′-biphenyldicarboxylate leads to the formation of Pd II 6 L 4 252; elongated ligands 253 ( Figure  66b) and 255 (Figure 66c) (254) and D 2h symmetries. 345 Later work showed that elongation of these tubelike structures through extension of the ligand with additional pyridine rings was not possible because of poor solubility. 346 This problem was circumvented by the design of hexapyridine ligand 257 ( Figure  66d) in which two terpyridine units are separated by a spacer. Importantly, the biphenyl spacer, as opposed to an alkyl moiety, ensured that the ligand remained linear instead of folding into a U-shaped conformation. Combination of 257 with cis-protected Pd II and the specially designed template molecule 258 allowed the construction of 3.5 nm long nanotube 259.

Pyrimidine versus Pyridine
The combination of pyridine donors with cis-protected squareplanar metal centers has yielded many new coordination cages, some of which are described in this review. Pyrimidine donors are also able to coordinate to M II centers, acting as 120°μ 2bridging ligands. Triangular hexadentate 1,3,5-tris(3,5-pyrimidyl)benzene (260) is designed to form structures in which the metal centers lie upon the edges of polyhedra as opposed to their vertices. 347 The combination of 260 with Pd II (en)(NO 3 ) 2 in aqueous solution yields enclosed Pd II 18 260 6 hexahedron 261 with a trigonal-bipyramidal structure consisting of six edge-sharing triangular panels with two metal centers on each edge ( Figure  67a). Subsequent work produced the similar Pd II 15 262 6 hexahedron 263 (Figure 67b) from modified ligand 262, in which one pyrimidine is replaced with a 3-pyridyl moiety. 348 Although 261 and 263 are of similar shape and size, the presence of open clefts at the apical nonbinding sites in 263 is hypothesized to allow easier access of guest molecules into the cavity for binding, in contrast to the more enclosed 261.
Architectures 261 and 263 demonstrate how the geometric constraints of a ligand can result in the formation of novel structures. Subsequent alterations can then be made that allow for a desired structure type to be maintained, but with altered properties. Further replacement of the pyrimidine sites of 262 with 3-pyridyl moieties yields different architectures of both higher and lower symmetries, including a tetrahedron and "open cones", depending on the conditions used. 349 Mukherjee et al. have also used ligands containing pyrimidine moieties to prepare complex architectures. The combination of  Figure 68) with Pd II in a 1:1 ratio produces structure 265. 351 Crystallography indicates that 265 is a discrete Pd II 24 264 24 complex wherein only four of the nitrogen atoms on each ligand bind to Pd II , leaving two uncoordinated nitrogen atoms on each ligand (highlighted in purple in Figure 68). The authors described this structure as a "pregnant molecular nanoball", consisting of a Pd II 12 "baby ball" within a larger Pd II 12 "mother ball". Interestingly, the "mother ball" stabilizes the internal, smaller, structure, as an analogous Pd II 12 L 24 cuboctahedral nanosphere does not form from the reaction between pyrimidine and Pd II . Duan and co-workers used a different f-block ion, cerium(IV), to construct a Ce IV 4 L 6 basketlike tetragon, using ditopic ligands consisting of two tridentate binding sites connected by a carbazole-based core. 353 Four Ce IV centers define the four Xu and Raymond exploited the variable coordination geometry of lanthanum to form a La III 8 L 8 structure with a square-antiprismatic geometry. 354 Within the structure, all of the lanthanum centers are nine-coordinate; however, two different coordination geometries are observed: distorted monocapped square-antiprismatic and distorted tricapped trigonal-prismatic. Similarly, Kawai et al. recently reported an octanuclear circular helicate with a D 4 -symmetric square-antiprismatic geometry that exhibited circularly polarized luminescence (CPL) activity. 355

Non-covalent Interactions and Steric Effects
Previous sections have alluded to the importance of steric control and favorable non-covalent interactions in driving the selective formation of particular structures. In this section we present a few key examples in which these often subtle effects exert a decisive influence on the self-assembly process.
Saalfrank et al. reported an early example wherein steric effects determine self-assembly outcomes. 356 The combination of rigid, threefold-symmetric, tris-bidentate pyrazolone-based ligand 268 with gallium(III) acetylacetonate in DMSO resulted in the serendipitous formation of Ga III 6 268 6 distorted trigonalantiprismatic cylinder 269 having D 3 symmetry, in which all six gallium centers have the same handedness ( Figure 70). Although this structure represents an initially surprising deviation from the expected M 4 L 4 tetrahedral assembly, molecular modeling indicated that the shorter metal−metal distance in a putative M 4 L 4 tetrahedron would cause an increase in unfavorable steric clashes. Further work by the same group utilized molecular modeling to clarify the steric preference for the formation of an Fe III 4 L 4 structure in the case of one ligand versus an Fe III 6 L 6 trigonal antiprism with a different ligand on the basis of favorable aromatic stacking interactions in the trigonal antiprism. 357 Clever and co-workers described the assembly of Pd II 2 270 3 bowl 271 (Figure 71) based on a Pd II 2 L 4 cage framework lacking a fourth ligand. 358 The formation of 271 is driven by the steric demands of its ligands. In the case of ligand 270, steric clash between hydrogen atoms near the coordinating nitrogen of the quinoline is alleviated in a Pd II 2 270 3 structure, stabilizing 271 with respect to a Pd II 2 270 4 structure. Upon prolonged heating of a solution of the Pd II 2 270 3 structure, partial conversion to the Pd II 2 270 4 structure was observed. However, this conversion could be prevented by the binding of C 60 in the open cavity of the bowl.
Bowl 271 can also act as a supramolecular protecting group. When a solution of [C 60 ⊂Pd II 2 270 3 Cl 2 ] 2+ is treated with 10 equiv of anthracene, the C 60 −anthracene monoadduct (C 60 Ac) is selectively formed without undergoing further reaction with anthracene. As shown in Figure 71, the bowl-like cavity of 271 encloses most of the surface of the C 60 guest, leaving only a small region available for reaction with anthracene. Finally, a pillshaped dimer can be formed by bridging the bowls of 2 equiv of 271 with a sterically undemanding terephthalate unit, 358 utilizing principles similar to those outlined in section 2.1. The example described above (Figure 71) demonstrates that introducing steric bulk close to the coordinating sites of ligands can increase their propensity to form more complex structures.
Mukherjee and co-workers reported the reaction of (1,1′bis(diphenylphosphino)ferrocene)platinum(II) with 5,10,15,20-tetrakis(4-pyridyl)porphyrin (272) ligands to form open hexagonal box 273 (Figure 72), as opposed to a more symmetric cubic architecture. 359 The bulky ferrocene-derived diphosphine ligand disfavors cube formation and allows formation of hexagonal box 273 with an internal cavity of 43 550 Å 3 . The formation of 273 depends on the introduction of steric bulk in peripheral coordinating ligands rather than directly affecting ligand−metal binding. This unusual self-assembled   Liu and co-workers built metal−organic architectures with complex structures using Eu III . The ditopic ligands 274, 276, and 278, each with two pyridine-2,6-dicarboxamide tridentate chelating sites connected by a 1,1′-bi-2-naphthol-derived core, produce architectures 275, 277, and 279, respectively ( Figure  73). 360 The key difference between the three ligands is the amount of steric bulk in proximity to the tridentate binding sites. All of the Eu III centers within the three structures are ninecoordinate. In 275, each of the six Eu III centers is chelated by three pyridine-2,6-dicarboxamide moieties, resulting in a structure with a twisted triangular-prismatic geometry ( Figure  73a). We also utilized steric effects and non-covalent interactions to drive the formation of barrel-like prismatic structures instead of simple tetrahedra. 229 Enforcing mer selectivity at the metal centers on the basis of reduced steric crowding and increased interligand aromatic stacking interactions enabled the formation of M 8 L 12 , M 10 L 15 , and M 12 L 18 prismatic barrel structures using fluorinated ligands. We hypothesize that the presence of favorable edge-to-face aromatic interactions between the triazine and phenanthroline moieties of neighboring ligands contributed to the stabilization of an unusual M 6 L 4 S 4symmetric scalenohedron over a higher-symmetry pseudooctahedral structure. 361 Finally, with Siegel and Baldridge we reported the assembly of an S 10 -symmetric, fivefold-interlocked [2]catenane from Cu I , a corannulene-based pentaaniline, and 2formyl-6-methylpyridine. 362 DFT calculations indicated that interligand aromatic interactions between corannulene units are a key driving force for the interlocking of the two fivefoldsymmetric cages.
This section highlights the critical role of geometric and steric factors, as well as the enhancement of non-covalent interactions, in controlling the final structure observed in self-assembly processes. Although a sizable amount of work has already been conducted in this area, these examples also illustrate that much space remains to be explored. Fujita's contributions, especially those involving the use of bis-monodentate ligands with Pd II metal centers, may inspire similar systematic studies focusing on small changes to one geometric feature of other classes of ligand, thereby leading to the discovery of general design principles for complex architectures.

CONCLUSION AND OUTLOOK
As we have detailed in this review, recent years have seen the rapid development of many new approaches to the construction of metal−organic structures beyond the Platonic solids. The advent of new applications for supramolecular cages, including in catalysis, 363−369 sensing, 370−372 molecular separations, 373 and biology, 374 has provided strong motivation for this development.   Chemical Reviews pubs.acs.org/CR Review As detailed in section 2, incorporating multiple different building blocks into the same structure is an effective way of increasing complexity. However, ensuring that mixed-component structures are formed selectively and that narcissistic sorting is prevented remains challenging. The six methods we outlined for driving heteroleptic architecture formation are variations on the theme of careful ligand selection and pairing. Recent efforts have demonstrated the reliability of heteroleptic approaches for the assembly of coordination cages, producing t a r g e t e d s t r u c t u r e s a c r o s s a r a n g e o f l i g a n d classes. 62,73,82,83,87,89,90,96,107,128,375 Similarly, as noted in section 6, an understanding of ligand coordination preferences can now allow the rational design of new heterometallic architectures. Employing metal ions that form coordination bonds to ligands with differing kinetic lability also provides a useful method of design. However, other methods of producing multimetallic architectures have been less thoroughly investigated, providing scope for future enquiry.
The principles of ligand flexibility, solvent effects, and templation detailed in sections 4 and 5 are well-established, and many early examples of complex architectures depend on these approaches. They can be unpredictable, however, and although serendipitous results are plentiful, targeted design from first principles remains a challenge. The factors that drive the formation of particular structures can be difficult to decipher, even after discovery. Key exceptions include groundbreaking work by Ward et al. involving thorough investigations of the selfassembly processes of flexible di-and tritopic pyrazolylpyridinebased ligands and octahedral metal centers, which led to several new classes of structures. 186−196 The reduced-symmetry-ligand and geometry-analysis approaches of sections 3 and 7 have benefited greatly from recent advances. As with heteroleptic approaches, it appears intuitive that reducing the symmetry of a ligand or building block should result in a self-assembled structure of reduced symmetry or increased complexity. However, experience has shown that it can be challenging to obtain single structures as opposed to intractable mixtures. Recent work has nonetheless clarified the circumstances under which a reduction in the symmetry of ligands can result in the formation of a single, complex architecture as opposed to many of them.
Finally, we note that computational methods, including evolutionary algorithms 376 and machine learning, 377 are playing an increasing role in the discovery of new supramolecular cage structures 378−380 and the rationalization of their applications, such as catalytic activity. 381 The recent implementation of highthroughput synthetic screening using automation has also vastly increased the capacity for rapid exploration of large chemical spaces. 382−384 The advent of artificial intelligence, in particular machine learning, and automated synthetic methods may thus play a key role in the structural prediction and subsequent synthesis of a broad range of low-symmetry metal−organic polyhedral capsules.

AUTHOR INFORMATION
Author Contributions † C.T.M. and J.A.D. contributed equally.

Notes
The authors declare no competing financial interest.